2 ||M|| This file is part of HELM, an Hypertextual, Electronic
3 ||A|| Library of Mathematics, developed at the Computer Science
4 ||T|| Department of the University of Bologna, Italy.
8 \ / This file is distributed under the terms of the
9 \ / GNU General Public License Version 2
10 V_______________________________________________________________ *)
12 include "basics/types.ma".
13 include "arithmetics/nat.ma".
15 inductive list (A:Type[0]) : Type[0] :=
17 | cons: A -> list A -> list A.
19 notation "hvbox(hd break :: tl)"
20 right associative with precedence 47
23 notation "[ list0 x sep ; ]"
24 non associative with precedence 90
25 for ${fold right @'nil rec acc @{'cons $x $acc}}.
27 notation "hvbox(l1 break @ l2)"
28 right associative with precedence 47
29 for @{'append $l1 $l2 }.
31 interpretation "nil" 'nil = (nil ?).
32 interpretation "cons" 'cons hd tl = (cons ? hd tl).
34 definition not_nil: ∀A:Type[0].list A → Prop ≝
35 λA.λl.match l with [ nil ⇒ True | cons hd tl ⇒ False ].
38 ∀A:Type[0].∀l:list A.∀a:A. a::l ≠ [].
39 #A #l #a @nmk #Heq (change with (not_nil ? (a::l))) >Heq //
43 let rec id_list A (l: list A) on l :=
46 | (cons hd tl) => hd :: id_list A tl ]. *)
48 let rec append A (l1: list A) l2 on l1 ≝
51 | cons hd tl ⇒ hd :: append A tl l2 ].
53 definition hd ≝ λA.λl: list A.λd:A.
54 match l with [ nil ⇒ d | cons a _ ⇒ a].
56 definition tail ≝ λA.λl: list A.
57 match l with [ nil ⇒ [] | cons hd tl ⇒ tl].
59 interpretation "append" 'append l1 l2 = (append ? l1 l2).
61 theorem append_nil: ∀A.∀l:list A.l @ [] = l.
62 #A #l (elim l) normalize // qed.
64 theorem associative_append:
65 ∀A.associative (list A) (append A).
66 #A #l1 #l2 #l3 (elim l1) normalize // qed.
69 ntheorem cons_append_commute:
70 ∀A:Type.∀l1,l2:list A.∀a:A.
71 a :: (l1 @ l2) = (a :: l1) @ l2.
74 theorem append_cons:∀A.∀a:A.∀l,l1.l@(a::l1)=(l@[a])@l1.
75 #A #a #l #l1 >associative_append // qed.
77 theorem nil_append_elim: ∀A.∀l1,l2: list A.∀P:?→?→Prop.
78 l1@l2=[] → P (nil A) (nil A) → P l1 l2.
79 #A #l1 #l2 #P (cases l1) normalize //
83 theorem nil_to_nil: ∀A.∀l1,l2:list A.
84 l1@l2 = [] → l1 = [] ∧ l2 = [].
85 #A #l1 #l2 #isnil @(nil_append_elim A l1 l2) /2/
88 (**************************** iterators ******************************)
90 let rec map (A,B:Type[0]) (f: A → B) (l:list A) on l: list B ≝
91 match l with [ nil ⇒ nil ? | cons x tl ⇒ f x :: (map A B f tl)].
93 lemma map_append : ∀A,B,f,l1,l2.
94 (map A B f l1) @ (map A B f l2) = map A B f (l1@l2).
97 | #h #t #IH #l2 normalize //
100 let rec foldr (A,B:Type[0]) (f:A → B → B) (b:B) (l:list A) on l :B ≝
101 match l with [ nil ⇒ b | cons a l ⇒ f a (foldr A B f b l)].
105 foldr T (list T) (λx,l0.if p x then x::l0 else l0) (nil T).
107 (* compose f [a1;...;an] [b1;...;bm] =
108 [f a1 b1; ... ;f an b1; ... ;f a1 bm; f an bm] *)
110 definition compose ≝ λA,B,C.λf:A→B→C.λl1,l2.
111 foldr ?? (λi,acc.(map ?? (f i) l2)@acc) [ ] l1.
113 lemma filter_true : ∀A,l,a,p. p a = true →
114 filter A p (a::l) = a :: filter A p l.
115 #A #l #a #p #pa (elim l) normalize >pa normalize // qed.
117 lemma filter_false : ∀A,l,a,p. p a = false →
118 filter A p (a::l) = filter A p l.
119 #A #l #a #p #pa (elim l) normalize >pa normalize // qed.
121 theorem eq_map : ∀A,B,f,g,l. (∀x.f x = g x) → map A B f l = map A B g l.
122 #A #B #f #g #l #eqfg (elim l) normalize // qed.
124 let rec dprodl (A:Type[0]) (f:A→Type[0]) (l1:list A) (g:(∀a:A.list (f a))) on l1 ≝
127 | cons a tl ⇒ (map ??(mk_Sig ?? a) (g a)) @ dprodl A f tl g
130 (**************************** length ******************************)
132 let rec length (A:Type[0]) (l:list A) on l ≝
135 | cons a tl ⇒ S (length A tl)].
137 notation "|M|" non associative with precedence 60 for @{'norm $M}.
138 interpretation "norm" 'norm l = (length ? l).
140 lemma length_append: ∀A.∀l1,l2:list A.
142 #A #l1 elim l1 // normalize /2/
145 (****************************** nth ********************************)
146 let rec nth n (A:Type[0]) (l:list A) (d:A) ≝
149 |S m ⇒ nth m A (tail A l) d].
151 lemma nth_nil: ∀A,a,i. nth i A ([]) a = a.
152 #A #a #i elim i normalize //
155 (****************************** nth_opt ********************************)
156 let rec nth_opt (A:Type[0]) (n:nat) (l:list A) on l : option A ≝
159 | cons h t ⇒ match n with [ O ⇒ Some ? h | S m ⇒ nth_opt A m t ]
162 (**************************** All *******************************)
164 let rec All (A:Type[0]) (P:A → Prop) (l:list A) on l : Prop ≝
167 | cons h t ⇒ P h ∧ All A P t
170 lemma All_mp : ∀A,P,Q. (∀a.P a → Q a) → ∀l. All A P l → All A Q l.
171 #A #P #Q #H #l elim l normalize //
175 lemma All_nth : ∀A,P,n,l.
178 nth_opt A n l = Some A a →
181 [ * [ #_ #a #E whd in E:(??%?); destruct
182 | #hd #tl * #H #_ #a #E whd in E:(??%?); destruct @H
185 [ #_ #a #E whd in E:(??%?); destruct
186 | #hd #tl * #_ whd in ⊢ (? → ∀_.??%? → ?); @IH
190 (**************************** Exists *******************************)
192 let rec Exists (A:Type[0]) (P:A → Prop) (l:list A) on l : Prop ≝
195 | cons h t ⇒ (P h) ∨ (Exists A P t)
198 lemma Exists_append : ∀A,P,l1,l2.
199 Exists A P (l1 @ l2) → Exists A P l1 ∨ Exists A P l2.
204 | #H cases (IH l2 H) /3/
208 lemma Exists_append_l : ∀A,P,l1,l2.
209 Exists A P l1 → Exists A P (l1@l2).
210 #A #P #l1 #l2 elim l1
218 lemma Exists_append_r : ∀A,P,l1,l2.
219 Exists A P l2 → Exists A P (l1@l2).
220 #A #P #l1 #l2 elim l1
222 | #h #t #IH #H %2 @IH @H
225 lemma Exists_add : ∀A,P,l1,x,l2. Exists A P (l1@l2) → Exists A P (l1@x::l2).
226 #A #P #l1 #x #l2 elim l1
228 | #h #t #IH normalize * [ #H %1 @H | #H %2 @IH @H ]
231 lemma Exists_mid : ∀A,P,l1,x,l2. P x → Exists A P (l1@x::l2).
232 #A #P #l1 #x #l2 #H elim l1
237 lemma Exists_map : ∀A,B,P,Q,f,l.
240 Exists B Q (map A B f l).
241 #A #B #P #Q #f #l elim l //
242 #h #t #IH * [ #H #F %1 @F @H | #H #F %2 @IH [ @H | @F ] ] qed.
244 lemma Exists_All : ∀A,P,Q,l.
248 #A #P #Q #l elim l [ * | #hd #tl #IH * [ #H1 * #H2 #_ %{hd} /2/ | #H1 * #_ #H2 @IH // ]
251 (**************************** fold *******************************)
253 let rec fold (A,B:Type[0]) (op:B → B → B) (b:B) (p:A→bool) (f:A→B) (l:list A) on l :B ≝
257 if p a then op (f a) (fold A B op b p f l)
258 else fold A B op b p f l].
260 notation "\fold [ op , nil ]_{ ident i ∈ l | p} f"
262 for @{'fold $op $nil (λ${ident i}. $p) (λ${ident i}. $f) $l}.
264 notation "\fold [ op , nil ]_{ident i ∈ l } f"
266 for @{'fold $op $nil (λ${ident i}.true) (λ${ident i}. $f) $l}.
268 interpretation "\fold" 'fold op nil p f l = (fold ? ? op nil p f l).
271 ∀A,B.∀a:A.∀l.∀p.∀op:B→B→B.∀nil.∀f:A→B. p a = true →
272 \fold[op,nil]_{i ∈ a::l| p i} (f i) =
273 op (f a) \fold[op,nil]_{i ∈ l| p i} (f i).
274 #A #B #a #l #p #op #nil #f #pa normalize >pa // qed.
277 ∀A,B.∀a:A.∀l.∀p.∀op:B→B→B.∀nil.∀f.
278 p a = false → \fold[op,nil]_{i ∈ a::l| p i} (f i) =
279 \fold[op,nil]_{i ∈ l| p i} (f i).
280 #A #B #a #l #p #op #nil #f #pa normalize >pa // qed.
283 ∀A,B.∀a:A.∀l.∀p.∀op:B→B→B.∀nil.∀f:A →B.
284 \fold[op,nil]_{i ∈ l| p i} (f i) =
285 \fold[op,nil]_{i ∈ (filter A p l)} (f i).
286 #A #B #a #l #p #op #nil #f elim l //
287 #a #tl #Hind cases(true_or_false (p a)) #pa
288 [ >filter_true // > fold_true // >fold_true //
289 | >filter_false // >fold_false // ]
292 record Aop (A:Type[0]) (nil:A) : Type[0] ≝
294 nill:∀a. op nil a = a;
295 nilr:∀a. op a nil = a;
296 assoc: ∀a,b,c.op a (op b c) = op (op a b) c
299 theorem fold_sum: ∀A,B. ∀I,J:list A.∀nil.∀op:Aop B nil.∀f.
300 op (\fold[op,nil]_{i∈I} (f i)) (\fold[op,nil]_{i∈J} (f i)) =
301 \fold[op,nil]_{i∈(I@J)} (f i).
302 #A #B #I #J #nil #op #f (elim I) normalize
303 [>nill //|#a #tl #Hind <assoc //]
306 (********************** lhd and ltl ******************************)
308 let rec lhd (A:Type[0]) (l:list A) n on n ≝ match n with
310 | S n ⇒ match l with [ nil ⇒ nil … | cons a l ⇒ a :: lhd A l n ]
313 let rec ltl (A:Type[0]) (l:list A) n on n ≝ match n with
315 | S n ⇒ ltl A (tail … l) n
318 lemma lhd_nil: ∀A,n. lhd A ([]) n = [].
322 lemma ltl_nil: ∀A,n. ltl A ([]) n = [].
323 #A #n elim n normalize //
326 lemma lhd_cons_ltl: ∀A,n,l. lhd A l n @ ltl A l n = l.
328 #n #IHn #l elim l normalize //
331 lemma length_ltl: ∀A,n,l. |ltl A l n| = |l| - n.
333 #n #IHn *; normalize /2/
336 (********************** find ******************************)
337 let rec find (A,B:Type[0]) (f:A → option B) (l:list A) on l : option B ≝
342 [ None ⇒ find A B f t
347 (********************** position_of ******************************)
348 let rec position_of_aux (A:Type[0]) (found: A → bool) (l:list A) (acc:nat) on l : option nat ≝
352 match found h with [true ⇒ Some … acc | false ⇒ position_of_aux … found t (S acc)]].
354 definition position_of: ∀A:Type[0]. (A → bool) → list A → option nat ≝
355 λA,found,l. position_of_aux A found l 0.
358 (********************** make_list ******************************)
359 let rec make_list (A:Type[0]) (a:A) (n:nat) on n : list A ≝
362 | S m ⇒ a::(make_list A a m)